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\Betreff[External Effects in MPSGE]{External Effects in MPSGE Models}

\begin{document}

  \begin{letter}{James Markusen \\
                 Department of Economics \\
                 University of Colorado \\
                 Boulder, CO}
    \opening{Hi Jim, }

We've reached the point in the MPSGEv2 project that we will be
incorporating the external effects.  I wanted to make sure that we get
started on the right track, so I figured that it make sense to spell
out the logic as precisely as possible.

Begin with a conventional CES production function:
$$y = f(x) = \phi \left( \sum_i \alpha_i x_i^\rho \right)^{1/\rho}$$

This might appear in MPSGE as:
\begin{verbatim}

                        $prod:Y  s:sigma
                           o:PY    q:y0
                           i:PX(i) q:x0(i)  p:p0(i)

\end{verbatim}

In the calibrated share format, we would write the production function
from the MPSGE model as:
$$y = f(x) = \bar y \left( \sum_i \theta_i \left( \frac{x_i}{\bar
x_i}\right)^\rho \right)^{1/\rho}$$
in which 
$$\theta_i = \frac{\bar p_i \bar x_i}{\sum_j \bar p_j \bar x_j}$$

We now want to modify the syntax to provide for the posibility of
external effects.  In the algebraic setting this would involve a
vector of productivity mulipliers, $\mu_i$, as in:
$$y = f(x) = \phi \left( \sum_i \alpha_i (\mu_i x_i)^\rho
\right)^{1/\rho}$$ 

Note that the idea here is that the scale and share parameters remain
unchanged, and all that the productivity multiplier does is enhance
the marginal product of the individual inputs.  In the MPSGE syntax,
we would represent this by an eXternal effects multiplier in the
\texttt{x:} field:

\begin{verbatim}

                $prod:Y  s:sigma
                   o:PY    q:y0
                   i:PX(i) q:x0(i)  p:p0(i)     x:MU(i)

\end{verbatim}
The syntax requires that a variable appearing in an \texttt{x:} field
\textit{must be an auxiliary variable}.


In the calibrated share form, we see that the resulting function would
be:
$$y = f(x) = \bar y \left( \sum_i \theta_i \left( \frac{x_i \mu_i}{\bar
x_i}\right)^\rho \right)^{1/\rho}$$

Logically, this is then equivalent to inverse adjustment of both the
reference quantity and reference price.  In other works,
representation of the external effects illustrated in the last model
with a fixed value multiplier, \texttt{mu0(i)}, we would need to write (in
the current version of MPSGE):

\begin{verbatim}
                        $prod:Y  s:sigma
                           o:PY    q:y0
                           i:PX(i) q:(x0(i)/mu0(i))  p:(p0(i)*mu0(i))

\end{verbatim}

I just wanted to run this by you before setting Florian to work on the
coding.  At Alex Meeraus' insistance, we will be making this
``backward compatible'', so it will eventually be included in the
current version of MPSGE as well as the new version which we are
working on.

    \closing{Mit freundlichen Gr\"ussen}
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  \end{letter}

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